Estimates of the Smoothness of Dyadic Orthogonal Wavelets of Daubechies Type

Suppose that $\omega(\varphi,\,\cdot\,)$ is the dyadic modulus of continuity of a compactly supported function $\varphi$ in $L^2(\mathbb R_+)$ satisfying a scaling equation with $2^n$ coefficients. Denote by $\alpha_\varphi$ the supremum for values of $\alpha>0$ such that the inequality $\omega(\varphi,2^{-j})\le C2^{-\alpha j}$ holds for all $j\in\mathbb N$. For the cases $n=3$ and $n=4$, we study the scaling functions $\varphi$ generating multiresolution analyses in $L^2(\mathbb R_+)$ and the exact values of $\alpha_\varphi$ are calculated for these functions. It is noted that the smoothness of the dyadic orthogonal wavelet in $L^2(\mathbb R_+)$ corresponding to the scaling function $\varphi$ coincides with $\alpha_\varphi$.